We point out that we only use manifolds without boundary or corners here.
For manifolds with boundary, part (1) of Proposition REF  fails—see, for instance, [1]} and [2]}.
Since each cartesian space \(c \in {{C}\mathrm {art}}\)  is diffeomorphic to \(\mathbb {R}^n\)  for some \(n \in \mathbb {N}_0\) , and since \(\mathbb {R}^n\)  is (isomorphic to) a CW complex for any \(n \in \mathbb {N}_0\) , it follows that \(\mathrm {D}\underline{c}\)  is cofibrant in \({\Delta {T}\mathrm {op}}\)  for every \(c \in {{C}\mathrm {art}}\) .
We have the following version of [3]}:
